Fracture Mechanics of Concrete Structures Proceedings FRAMCOS-3 Aedificatio Publishers, D Frei burg, Germany OF THE STANDARD

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1 Fracture Mechanics of Concrete Structures Proceedings FRAMCOS-3 Aedificatio Publishers, D Frei burg, Germany DETERMINATION PARAMETERS IN NOTCHED BEAMS OF THE STANDARD DOUBLE-K FRACTURE THREE-POINT BENDING S. Xu and H.W. Reinhardt Institute of Construction Materials, University of Stuttgart, Stuttgart, Germany Abstract In this paper, an attempt is made to determine the double-k fracture parameters Kicini and K1c un using three-point bending tests. Based on the linear asymptotic superposition assumption proposed by the authors the critical effective crack length ac; is analytically evaluated by inserting the critical crack mouth opening displacement CMODc and the maximum load Pmax, i.e. the secant compliance Cs, into the formula of LEFM. Using the analytical solution of a fictitious crack with cohesive forces in an infinite strip model the double-k fracture parameters K1cini and K1cun as well as the critical crack tip opening displacement CTODc were analytically determined. The experimental evidence showed that the double-k fracture parameters K1cini and K1c 0 n are size-independent and can be considered as the material parameters to describe the material properties of cracking initiation and unstable fracture in concrete structures. The testing methods required for the determination of the Kicini and Kicun is quite simple. A closed-loop testing system is not necessary. Key words: Double-K fracture parameters, standard tests, cracking initiation, unstable fracture, cohesive force, fictitious crack model. 431

2 1 Introduction Based on several fracture models which include the fictitious crack model (FCM) by Hillerborg et al. (1976), the size effect model(sem) by Bazant and Kazemi (1990), the two-parameter fracture model (TPFM) by Jenq and Shah(1985) and the effective crack model (ECM) by Karihaloo and Nallathambi ( 1986) and Swartz and Refai ( 1987), the corresponding methods to measure the fracture parameters, for examples, the fracture energy GF introduced in the FCM, the critical energy release rate Gr and the critical effective crack extension cr for the infinite specimen in the SEM, and the critical stress intensity factor Krc 5 and the critical crack tip opening displacement CTODc in the TPFM, were recommended by RILEM in 1985 and in 1990 respectively. To describe the different stages of crack propagation including crack initiation, stable crack propagation and unstable fracture a double-k fracture criterion for crack propagation in the quasi-brittle material like concrete was proposed by Xu and Reinhardt (1997a). In this paper an attempt is made to combine the cohesive force on the fictitious crack with the double-k crack propagation criterion based on stress intensity factor. As the result, a practical measuring method is proposed using standard three-point bending beams. 2 Evaluation of the Effective Crack Length In order to evaluate the effective crack length a linear asymptotic superposition assumption was proposed by Xu and Reinhardt ( l 997b) which takes the nonlinear part of the P-CMOD curve into account. According to the assumption, the effective crack length ac can be determined on the basis of linear elastic fracture mechanics (see Tada's Stress Analysis of Cracks Handbook, 1985). CMOD = 6 PSa D 2 BE (a) (1) for SID= 4, the function V 1 (a) is given as follows: V 1 (a) = a+3.87a -2.04a + 2 (1-a) (2) 432

3 where a=(a + H 0 )/(D+H 0 ), P = Load, S = specimen loading span, D = beam depth, B = beam width, H 0 = thickness of clip gauge holder. Young's modulus E can be calculated from the measured initial compliance Ci through a method proposed by Jenq and Shah (1985) using the equation (3): (3) where ao = (ao + H 0 )/(D + H 0 ) and ao =initial crack length On the other hand, according to the testing results of Karihaloo and Nallathambi (1991), Young"s modulus E from standard compressive cylinder tests can be used to predict the average length of the critical effective crack of a group of specimens. 3 The approaches to determine double-k parameters Kinik and Ku"1c Concrete as a quasi-brittle softening material shows three different situations of crack propagation: crack initiation, stable crack propagation and unstable fracture. The proposed double-k fracture parameters Kiniic and Kunic can be applied to these such three different situations. The criterion for unstable fracture is defined as the critical stress intensity factor Kunic So, for a three-point bending notched beam, the Kunic can be evaluated by inserting the maximum load P max and the critical effective crack length ac into the following expression (Tada, 1985): (4) where the geometry factor F 1 (a/d) depends on the ratio of span to depth of the beam and which is for S = 4D given as follows: (5) Theoretically speaking, the initiation toughness Kiniic is defined as the 433

4 initial cracking stress intensity factor created at the initial crack tip by the cracking load Pi. However, in practical experiments, to distinguish a sole initial cracking load Pi by various investigation approaches is not In ordinary tests, it is not convenient too. Therefore, another approach to determine the initiation toughness Kini1c is presented. The initiation toughness Kini1c, in fact, is the inherent toughness of a material. It implies that a crack does not propagate when the stress intensity factor at the initial crack tip is less than the inherent toughness, the initiation toughness Kiniic Due to the steady crack propagation, the toughness of a loaded body increases from the value of Kiniic to the one of Kunic The contribution due to cohesive forces is called Kele This leads to: Kini+ Kc =Kun le le le (6) UH' '~.,., critical situation at which the maximum external load is reached crack tip opening displacement arrives at its critical value CTODe,... L bending tests, the cohesive toughness Kele can be calculated a Green's function as follows: (7) when x = ac, u = 1; x = ao, u = aof ac, eq. (7) can be rewritten as (8) correspondingly, where 3.52(1-U) U +{l u 312 +O.S U} {l-(l-u v} (9) (l-v)312 (l-v)112 (l-u2)112 ) 434

5 The corresponding cohesive force distribution cr(x/ac) on the fictitious crack zone shown in equation (8) can be expressed as follows: At the integral boundary of eq. (8), the integration has a singularity. Numerical results of the integral can be gained by using Gauss Chebyshev guadrature. The crs(ctode) value in equation (10) can be determined by an expression proposed by Reinhardt et al. (1986) as follows: (J { w 3} w w 3 - = 1 +(c 1 -) exp(-c 2 -)--(1 +c 1 )exp(-c 2 ) fr Wo Wo Wo (11) where the coefficient c 1, c 2 are constants and the w 0 is the maximum crack opening width at the stress to be zero. Of course, for the aim of simplicity, a bilinear cr-w relation can be used too. Once, the CMODe is measured in the tests, the CTODe can be evaluated by the following expression (see Jenq and Shah (1985)): COD(x)=CMOD c (1-;) 2 [ 2]}1/2 { +( ;);-(;) (12) The detailed determination procedure contains the following steps: 1. According to the initial compliance Ci taken from the linear segment of the P-CMOD curve Young's modulus Eis calculated using eq. (3),.or E measured from compressive cylinder tests can be used. 2. Pmruo CMODe and E are inserted into eq. (1) to calculate the critical effective crack length ac. 3. Submitting Pmax and ac into eq. (4), Kunic can be obtained. 4. Using CMODe and ac CTODc is evaluated by eq. (12). Then, inserting CTODc into eq. (12), <J 5 (CTODc) in eq. (11) can be gained. 5. Carrying out a numerical scheme, the integral value Ke1c of eq. (7) is received. 6. Finally, Kun1c and Ke1c are inserted into eq. (6) and the initiation toughness Kiniic can be got too. 435

6 4 Experimental Validation The validation of the proposed method for determining the double-k fracture parameters is carried out on the data of Refai and Swartz (1987). It must be noted that both series B and C, before a beam was tested to failure the beams were precracked using strain control. The initial precrack lengths of ten specimens in series B and fourteen specimens in series C were directly measured using a dye-penetrant technique. Then the authors carried out a satisfactory regression to gain the regression expressions of the compliance calibration curve and the maximum load calibration curve. As a result, the lengths of initial precracks ai of other specimens were evaluated by the regression expression. In our evaluation the lengths of the initial precracked cracks ai, the initial compliance C each beam was carefully determined according to the P-CMOD curves presented in the report of Refai and Swartz (1987). Using the measured values of the initial compliance Ci the lengths of the initial precracked cracks ai can be calculated according to eq. (1). The related lengths a/d of the initial precracks to the depth evaluated by eq. (1) and the corresponding values of a/d obtained by the dye-penetrant technique and regression expressions in the report of Refai and Swartz (1987) are compared and plotted in Fig. 1 (a) and (b) for series B and C separately. It can be seen that the values of a/d evaluated by eq. (1) are in good agreement with those gained by the approaches used by Refai and Swartz (1987). The mean error is less than 2% compared with those measured by the dye-penetrant technique and is less than 1 % compared with those evaluated by the maximum load calibration method and less than 7% compared with those evaluated by the compliance calibration method by Refai and Swartz (1987) respectively for both series B and C. The corresponding coefficients of variation are less than 8%, 9% and 7%, respectively. Now, the above-mentioned procedures to determine the double-k fracture parameters Kiniic and Kunic will be used. In the calculation, for convenience, the cr-w relation shown in eq. (6) and corresponding c 1 = 3, c 2 = 7 and w 0 = 160 µm were used. All values evaluated are presented in Table 1 and Table 2 for series B and C, respectively. The average values of Kiniic and Kunic for series B are MPa m 112 and MPa m 112. These values for series C are MPa m 112 and MPa m 112 The coefficients of variation of the measured values for series B are and These values for series C are and This means that in the region of the tested specimen sizes, the 436

7 evaluated values of Kunrc and Kini le are size-independent. Table 1. The results of double-k fracture parameters Kiniic and Kunic determined for series B (S x D x B = 762 x 203 x 76 mm, Ho =3.2 mm, fc = 53.1 MPa, E = 38.4 GPa) Nos.of Cixlff 0 a/d Pmax CMODc CTODc ~ID ~c ~Ill ~m specs. (mm/n) (N) (µm) (µm) (MPam~ (MPam~ (MPam~ Bl B l B BS B BS B BIO Bll Bl Bl Bl Bl Bl? B B B B B B B B B B B B B B B B B B B B B B B B B B B Mean S.D C.V

8 ,, c m a; 0.8 -Cl)- a:... IX) >en.0,...,, 'N 0.6 I!! t:: :;, m en ;: «I (/) Cl) E ~ m Comparison for Series B 0.6 aifd by equation (1) (a) 0 + ai/d by CCM of Guo and Swartz(1984) A ai/d measured by dyepenetrant (Refai and Swartz) o ai/d by MLCM of Refai and Swartz(1987) 0.8 "Cl r: m "ii l> a:... co >- en.q,... "Cl - ::I I'll II) ;: I!! t! 0.6 CG Cl) a> E 0.4 ~ I'll Comparison for Series C.A. ai/d measured by dye-penetrant (Refai and Swartz, 1987) o ai/d by MLCM of Refai and Swartz(1987) aifd by equation (1) (b) Fig.I. The comparison between initial crack lengths evaluated by eq.(1) and those measured: (a) for series B; and (b) for series C beams. It can be also observed that the coefficients of variation of the measured values Kiniic and Kunic are in the same range as those for normal strength parameters of concrete like fb fc. For example, the coefficients of variation of the measured values of fb fc for the series B cylinder data are and But, the mean values of CTODc measured in the two series beams seem size-dependent. For the series B the mean value of CTODc is 17.4 µm and for series C it is µm. 438

9 Table 2. The results of Kiniic and Kunic determined for series C.( S x D x B = 1143 x 305 x 76 mm, H 0 =3.2 mm, fc= 54.4 MPa, E = 39.3 GPa) Nos.of CixlO a/d Pmax CMODc CTODc ajd K1cc K 1111 le specs. (mm!ny6 (N) (µm) (µm) (MPam 112 ) (MPamin) (MPa m 112 ) Cl C C C cs C C C C ClO Cll C C Cl Cl C C C C C C C C C C C C C C l.449 C C C C Mean S.D C.V Kun Jc 5 Conclusions An experimental method is proposed to determine double-k fracture parameters Kini 1c and Kunk on single size three-point bending notched beams. The initial compliance Ci and the secant compliance Cs is measured without unloading and reloading procedures. It only needs a monotonically increasing load on a beam until the is 439

10 reached and the measurement of the P-CMOD curve in the ascending branch. In common materials and structures laboratories, this testing method can be performed even without a closed-loop testing system. The experimental evidence shows that the double-k fracture parameters Kini le and Kun1c are size-independent and can be used to describe the material features of cracking initiation and unstable fracture of concrete structures. 6 References Bazant, Z.P. and Kazemi, M. T. (1990) Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. Int. J. Fracture, 44, Hillerborg, A. Modeer, M. and Petersson, P.E., (1976), Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. & Coner. Res. 6, Jenq, Y.S. and Shah, S.P.(1985) Two parameter fracture model for concrete. J. Engin. Mech. (ASCE), 111, Nallathambi, P. and Karihaloo, B.L. (1986) Determination of specimensize independent fracture toughness of plain concrete. Magz. of Coner. Res., 38, Karihaloo, B.L. and Nallathambi, P. (1991) Notched beam best: mode I fracture toughness, in Fracture Mechanics Test Methods for Concrete( eds S.P. Shah and A. Carpinteri), Chapman & Hall, London, Refai, T.M.E. and Swartz, S.E. (1987) Fracture behavior of concrete beams in three-point bending considering the influence of size effects. Report No.190, Engin. Exper. Station, Kansas State University. Reinhardt, H.W. et al. (1986) Tensile tests and failure analysis of concrete. J. Struct. Engin., 112, Tada, H., Paris, P.C. and R. Irwin, G. (1985) The Stress Analysis of Cracks Handbook. Paris Productions Incorporated, Missouri, USA. Xu, S. and Reinhardt, H.W. (1997a) Determination of double-k criterion for crack propagation in quasi-brittle materials, Part 1: Experimental investigation of crack propagation, Int. J. Fracture, submitted for publication. Xu, S. and Reinhardt, H.W. (1997b) Crack extension resistance and fracture properties of quasi-brittle materials like concrete based on the complete process of fracture, Int. J. Fracture, submitted for publication. 440